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Properties

Label 16T50
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $SD_{16}:C_2$

Related objects

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Group action invariants

Degree $n$:  $16$
Transitive number $t$:  $50$
Group:  $SD_{16}:C_2$
Parity:  $1$
Primitive:  no
Nilpotency class:  $3$
$|\Aut(F/K)|$:  $8$
Generators:  (1,8,2,7)(3,6,4,5)(9,12,10,11)(13,15,14,16), (9,10)(11,12)(13,14)(15,16), (1,13)(2,14)(3,11)(4,12)(5,15)(6,16)(7,10)(8,9)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$

Low degree siblings

16T32, 32T18

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,13,12,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 3,16, 2,10, 4,15)( 5,11, 7,13, 6,12, 8,14)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 4,15, 2,10, 3,16)( 5,11, 8,14, 6,12, 7,13)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3,14, 4,13)( 5,10, 6, 9)( 7,16, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,16)( 8,15)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [32, 44]
Character table:    
      2  5  4  5  4  4  3  3  3  3  3  3

        1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c
     2P 1a 1a 1a 2b 2b 2b 2b 4b 4b 2b 1a
     3P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c
     5P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c
     7P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1 -1  1 -1  1 -1  1 -1  1
X.3      1 -1  1 -1  1 -1  1  1 -1  1 -1
X.4      1 -1  1 -1  1  1 -1 -1  1  1 -1
X.5      1 -1  1 -1  1  1 -1  1 -1 -1  1
X.6      1  1  1  1  1 -1 -1 -1 -1  1  1
X.7      1  1  1  1  1 -1 -1  1  1 -1 -1
X.8      1  1  1  1  1  1  1 -1 -1 -1 -1
X.9      2  2  2 -2 -2  .  .  .  .  .  .
X.10     2 -2  2  2 -2  .  .  .  .  .  .
X.11     4  . -4  .  .  .  .  .  .  .  .